Let
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
positive integers
![a_1, \dots , a_m](/media/m/b/2/f/b2f6acd654e7f283e8194dd0fb175892.png)
be given. Prove that there exist fewer than
![2^m](/media/m/9/f/d/9fd21b1dfc0b1a705cfdb123862acd4f.png)
positive integers
![b_1, \dots , b_n](/media/m/7/b/b/7bb813097824c667e7985526f10f3f73.png)
such that all sums of distinct
![b_k](/media/m/1/e/1/1e1016cbade6ef5706a4e3f9c56841bc.png)
’s are distinct and all
![a_i \ (i \leq m)](/media/m/b/b/8/bb889c6aeb0764fde3a4c8d66b7d5387.png)
occur among them.
%V0
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.